ON OPIALS INEQUALITY FOR F(N)

We prove inequalities of the type integral-0/h\f(i)(x)f(j)(x)\dx less-than-or-equal-to C(n, i, j, p)h2n - i - j + 1 - 2/p(integral-0/h\f(n)(x)\p dx)2/p when f(0) = f'(0) = ... = f(n - 1)(0) = 0. We assume that f(n-1) is absolutely continuous and f(n) is-an-element-of L(p)(0, h), with p greater-than-or-equal-to 1, n greater-than-or-equal-to 2, and 0 less-than-or-equal-to i less-than-or-equal-to j less-than-or-equal-to n - 1.